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Math Help - how to do this ?

  1. #1
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    how to do this ?

    Determine whether the function satisfies the hypotheses of the mean value theorem for the given interval

    f(x) = tan^(-1) x , [-1,1]
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  2. #2
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    Quote Originally Posted by yakkow
    Determine whether the function satisfies the hypotheses of the mean value theorem for the given interval

    f(x) = tan^(-1) x , [-1,1]
    Is is countinous on the closed interval [-1,1]--->Yes
    Is is differenciable on the open interval (-1,1)---->Yes
    Conditions satisfied.
    ---
    To find such a number you need.

    f'(c)=\frac{f(b)-f(a)}{b-a}
    Since,
    f'(x)=\frac{1}{1+x^2}
    We have,
    f'(c)=\frac{1}{1+c^2}
    Also, f(b)=f(1)=\pi/4, f(a)=f(-1)=-\pi/4
    Thus,
    \frac{1}{1+c^2}=\frac{\pi/4-(-\pi/4)}{1-(-1)}=\frac{\pi/2}{2}=\frac{\pi}{4}
    Thus,
    1+c^2=\frac{4}{\pi}
    Thus,
    c^2=\frac{4-\pi}{\pi}
    Thus,
    c=\pm \sqrt{\frac{4-\pi}{\pi}
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